Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. A minimum product spanning tree for a weighted, connected and undirected graph is a spanning tree with weight product less than or equal to the weight product of every other spanning tree. The weight product of a spanning tree is the product of weights corresponding to each edge of the spanning tree. All weights of the given graph will be positive for simplicity.
Minimum Product that we can obtain is 180 for above graph by choosing edges 0-1, 1-2, 0-3 and 1-4
This problem can be solved using standard minimum spanning tree algorithms like krushkal and prim’s algorithm, but we need to modify our graph to use these algorithms. Minimum spanning tree algorithms tries to minimize total sum of weights, here we need to minimize total product of weights. We can use property of logarithms to overcome this problem.
As we know,
log(w1* w2 * w3 * …. * wN) = log(w1) + log(w2) + log(w3) ….. + log(wN)
We can replace each weight of graph by its log value, then we apply any minimum spanning tree algorithm which will try to minimize sum of log(wi) which in-turn minimizes weight product.
For example graph, steps are shown in below diagram,
In below code first we have constructed the log graph from given input graph, then that graph is given as input to prim’s MST algorithm, which will minimize the total sum of weights of tree. Since weight of modified graph are logarithms of actual input graph, we actually minimize the product of weights of spanning tree.
Edge Weight 0 - 1 2 1 - 2 3 0 - 3 6 1 - 4 5 Minimum Obtainable product is 180
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